Volume 24, Issue 1

The possible densities of graphs in which each edge lies in at least two maximal cliques are found and the graphs of maximal densities are described.

Let Γ be a connected edge-regular graph with parameters (v, k, λ), b 1 = k−λ−1. It is well known that if b 1 = 1, then Γ is a polygon or a complete multipartite graph with colour classes of order 2. The classification of graphs with b 1 ≤ 4 is available. Even in the case b 1 = 5 the study of graphs offers great difficulties. However, the situation is much easier for strongly regular graphs. In the present paper we classify strongly regular graphs with b 1 < 26.

The function algebra of first-degree polynomials over ring ℤ is analysed. The algorithm for completeness recognition in this algebra is described, its time and space complexity are estimated. The algorithm is applied to some other problems: recognition properties of polynomials and completeness of their special sets.

The paper is concerned with methods for calculation of the Hadamard product of rational functions and applications of Hadamard products to probabilistic problems. Explicit formulas for the Hadamard products of rational functions are given in the form of the ratio of determinants of matrices whose entries depend on the coefficients of the numerator and denominator of fractions defining these functions. An application to calculation of distributions of statistics relating to oscillating random walks with integer steps is considered.

Let F q be a finite field of characteristic p with q = p v elements, g a primitive element, a ≠ 0 an arbitrary element of F q , and x = log g a the discrete logarithm of a to the base g. In this paper we consider the discrete logarithm problem in the case when q ≡ 1(mod 4) and put forward a deterministic algorithm computing the first k ≦ c log n digits x 0 , x 1 , . . . , x k , k < n, in the binary expansion x = x 0 +x 1 2+x 2 2 2 + ··· +x n 2 n of x in a polynomial time.

Discrete spaces of elementary events with families of probabilities depending on a parameter ë are considered. We introduce the notion of events independent relative to a family of distributions and the notion of families of distributions without independent events. A simple necessary condition of the existence of events independent relative to the family of power series distributions is given. It permits to investigate a question on the existence of events independent relative to these families of distributions. Examples of families of distributions having independent events, families without independent events, and also the families not belonging to both classes are given.